Mathematics for engineers

Mathematics for Engineers

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Mathematics for

Engineers

Fourth Edition

Anthony Croft

Loughborough University

Robert Davison

Harlow, England • London • New York • Boston • San Francisco • Toronto • Sydney • Singapore • Hong Kong

Tokyo • Seoul • Taipei • New Delhi • Cape Town • Madrid • Mexico City • Amsterdam • Munich • Paris • Milan

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PEARSON EDUCATION LIMITED

Edinburgh Gate

Harlow CM20 2JE

United Kingdom

Tel: +44(0)1279 623623

Web: www.pearson.com.uk

First published 1998 (print)

Second edition published 2004 (print)

Third edition published 2008 (print)

Fourth edition published 2015 (print and electronic)

© Pearson Education Limited 1998, 2004, 2008, (print)

© Pearson Education Limited 2015 (print and electronic)

The rights of Anthony Croft and Robert Davison to be identified as authors of this work have been

asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

The print publication is protected by copyright. Prior to any prohibited reproduction, storage in a

retrieval system, distribution or transmission in any form or by any means, electronic, mechanical,

recording or otherwise, permission should be obtained from the publisher or, where applicable, a

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Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, London EC1N 8TS.

The ePublication is protected by copyright and must not be copied, reproduced, transferred,

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Pearson Education is not responsible for the content of third-party internet sites.

ISBN: 978-1-292-06593-9 (print)

978-1-292-07775-8 (PDF)

978-1-292-07774-1 (eText)

British Library Cataloguing-in-Publication Data

A catalogue record for the print edition is available from the British Library

Library of Congress Cataloging-in-Publication Data

A catalog record for the print edition is available from the Library of Congress

10 9 8 7 6 5 4 3 2 1

19 18 17 16 15

Cover: Dubai Meydan bridge, ALMSAEED/Getty Images

Print edition typeset in 10/12 Times by 73

Printed in Slovakia by Neografia

NOTE THAT ANY PAGE CROSS REFERENCES REFER TO THE PRINT EDITION

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To Kate and Harvey (AC)

To Kathy (RD)

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Brief contents

Contents ix

Publisher’s acknowledgements xv

Preface xvi

Using mathematical software packages xx

1 Arithmetic 1

2 Fractions 16

3 Decimal numbers 33

4 Percentage and ratio 43

5 Basic algebra 55

6 Functions and mathematical models 134

7 Polynomial equations, inequalities,

partial fractions and proportionality 208

8 Logarithms and exponentials 279

9 Trigonometry 325

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10 Further trigonometry 391

11 Complex numbers 440

12 Matrices and determinants 499

13 Using matrices and determinants to solve equations 576

14 Vectors 643

15 Differentiation 710

16 Techniques and applications of differentiation 735

17 Integration 790

18 Applications of integration 859

19 Sequences and series 907

20 Differential equations 937

21 Functions of more than one variable

and partial differentiation 1008

22 The Laplace transform 1040

23 Statistics and probability 1073

24 An introduction to Fourier series

and the Fourier transform 1157

Typical examination papers 1178

Appendix 1: SI units and prefixes 1184

Index 1185

viii Brief contents

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Contents

Publisher’s acknowledgements xv

Preface xvi

Using mathematical software packages xx

1 Arithmetic 1

Block 1 Operations on numbers 3

Block 2 Prime numbers and prime factorisation 10

End of chapter exercises 15

2 Fractions 16

Block 1 Introducing fractions 18

Block 2 Operations on fractions 23

End of chapter exercises 31

3 Decimal numbers 33

Block 1 Introduction to decimal numbers 35

Block 2 Significant figures 40

End of chapter exercises 41

4 Percentage and ratio 43

Block 1 Percentage 45

Block 2 Ratio 49

End of chapter exercises 54

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5 Basic algebra 55

Block 1 Mathematical notation and symbols 57

Block 2 Indices 70

Block 3 Simplification by collecting like terms 86

Block 4 Removing brackets 89

Block 5 Factorisation 97

Block 6 Arithmetic of algebraic fractions 104

Block 7 Formulae and transposition 117

End of chapter exercises 131

6 Functions and mathematical models 134

Block 1 Basic concepts of functions 136

Block 2 The graph of a function 145

Block 3 Composition of functions 153

Block 4 One-to-one functions and inverse functions 156

Block 5 Parametric representation of a function 163

Block 6 Describing functions 166

Block 7 The straight line 175

Block 8 Common engineering functions 189

End of chapter exercises 205

7 Polynomial equations, inequalities,

partial fractions and proportionality 208

Block 1 Solving linear equations 211

Block 2 Solving quadratic equations 221

Block 3 Factorising polynomial expressions and solving

polynomial equations 234

Block 4 Solving simultaneous equations 243

Block 5 Solution of inequalities 252

Block 6 Partial fractions 261

Block 7 Proportionality 272

End of chapter exercises 276

8 Logarithms and exponentials 279

Block 1 The exponential function 281

Block 2 Logarithms and their laws 296

Block 3 Solving equations involving logarithms and exponentials 306

Block 4 Applications of logarithms 311

End of chapter exercises 322

9 Trigonometry 325

Block 1 Angles 327

Block 2 The trigonometrical ratios 331

Block 3 The trigonometrical ratios in all quadrants 342

x Contents

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Block 4 Trigonometrical functions and their graphs 350

Block 5 Trigonometrical identities 362

Block 6 Trigonometrical equations 367

Block 7 Engineering waves 376

End of chapter exercises 389

10 Further trigonometry 391

Block 1 Pythagoras’s theorem and the solution

of right-angled triangles 393

Block 2 Solving triangles using the sine rule 403

Block 3 Solving triangles using the cosine rule 409

Block 4 Surveying 414

Block 5 Resolution and resultant of forces 425

End of chapter exercises 437

11 Complex numbers 440

Block 1 Arithmetic of complex numbers 442

Block 2 The Argand diagram and polar form of a complex number 453

Block 3 The exponential form of a complex number 468

Block 4 De Moivre’s theorem 474

Block 5 Solving equations and finding roots of complex numbers 482

Block 6 Phasors 490

End of chapter exercises 496

12 Matrices and determinants 499

Block 1 Introduction to matrices 501

Block 2 Multiplication of matrices 511

Block 3 Determinants 520

Block 4 The inverse of a matrix 539

Block 5 Computer graphics 548

End of chapter exercises 571

13 Using matrices and determinants

to solve equations 576

Block 1 Cramer’s rule 579

Block 2 Using the inverse matrix to solve simultaneous equations 583

Block 3 Gaussian elimination 591

Block 4 Eigenvalues and eigenvectors 604

Block 5 Iterative techniques 620

Block 6 Electrical networks 629

End of chapter exercises 639

14 Vectors 643

Block 1 Basic concepts of vectors 645

Block 2 Cartesian components of vectors 659

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Block 3 The scalar product, or dot product 677

Block 4 The vector product, or cross product 689

Block 5 The vector equation of a line and a plane 700

End of chapter exercises 708

15 Differentiation 710

Block 1 Interpretation of a derivative 712

Block 2 Using a table of derivatives 721

Block 3 Higher derivatives 730

End of chapter exercises 733

16 Techniques and applications of differentiation 735

Block 1 The product rule and the quotient rule 737

Block 2 The chain rule 743

Block 3 Implicit differentiation 749

Block 4 Parametric differentiation 755

Block 5 Logarithmic differentiation 759

Block 6 Tangents and normals 763

Block 7 Maximum and minimum values of a function 773

End of chapter exercises 787

17 Integration 790

Block 1 Integration as differentiation in reverse 792

Block 2 Definite integrals 804

Block 3 The area bounded by a curve 811

Block 4 Computational approaches to integration 821

Block 5 Integration by parts 831

Block 6 Integration by substitution 838

Block 7 Integration using partial fractions 849

Block 8 Integration of trigonometrical functions 852

End of chapter exercises 856

18 Applications of integration 859

Block 1 Integration as the limit of a sum 861

Block 2 Volumes of revolution 867

Block 3 Calculating centres of mass 874

Block 4 Moment of inertia 887

Block 5 The length of a curve and the area of a surface

of revolution 893

Block 6 The mean value and root-mean-square

value of a function 899

End of chapter exercises 906

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19 Sequences and series 907

Block 1 Sequences and series 909

Block 2 Sums of whole numbers, their squares and cubes 920

Block 3 Pascal’s triangle and the binomial theorem 923

Block 4 Taylor series and Maclaurin series 929

End of chapter exercises 935

20 Differential equations 937

Block 1 Basic concepts of differential equations 940

Block 2 Separation of variables 955

Block 3 Solving first-order linear equations using an integrating factor 963

Block 4 Computational approaches to differential equations 971

Block 5 Second-order linear constant-coefficient equations I 981

Block 6 Second-order linear constant-coefficient equations II 994

End of chapter exercises 1006

21 Functions of more than one variable

and partial differentiation 1008

Block 1 Functions of two independent variables, and their graphs 1010

Block 2 Partial differentiation 1020

Block 3 Higher-order derivatives 1029

Block 4 Stationary values of a function of two variables 1033

End of chapter exercises 1038

22 The Laplace transform 1040

Block 1 The Laplace transform 1042

Block 2 The inverse Laplace transform 1051

Block 3 Solving differential equations using

the Laplace transform 1060

End of chapter exercises 1070

23 Statistics and probability 1073

Block 1 Data 1075

Block 2 Data averages 1077

Block 3 Variation of data 1085

Block 4 Elementary probability 1090

Block 5 Laws of probability 1099

Block 6 Probability distributions 1113

Block 7 The binomial distribution 1121

Block 8 The Poisson distribution 1129

Block 9 The normal distribution 1138

End of chapter exercises 1154

Contents xiii

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24 An introduction to Fourier series

and the Fourier transform 1157

Block 1 Periodic waveforms and their Fourier representation 1159

Block 2 Introducing the Fourier transform 1169

End of chapter exercises 1176

Typical examination papers 1178

Appendix 1: SI units and prefixes 1184

Index 1185

xiv Contents

Companion Website

For open-access student resources specifically

written to complement this textbook and support

your learning, please visit www.pearsoned.co.uk/croft

Lecturer Resources

For password-protected online resources tailored to support

the use of this textbook in teaching, please visit

www.pearsoned.co.uk/croft

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